THE VALUE OF SCIENCE 401 



after all to proclaim that they deceive themselves, that their straight 

 is not the true straight, if we still are unwilling to admit that such an 

 affirmation has no meaning, at least we must confess that these people 

 have no means whatever of recognizing their error. 



2. Qualitative Geometry 



All that is relatively easy to understand, and I have already so 

 often repeated it that I think it needless to expatiate further on the 

 matter. Euclidean space is not a form imposed upon our sensibility, 

 since we can imagine non-Euclidean space; but the two spaces, 

 Euclidean and non-Euclidean, have a common basis, that amorphous 

 continuum of which I spoke in the beginning. From this continuum 

 we can get either Euclidean space or Lobachevskian space, just as we 

 can, by tracing upon it a proper graduation, transform an ungraduated 

 thermometer into a Fahrenheit or a Eeaumur thermometer. 



And then comes a question : Is not this amorphous continuum that 

 our analysis has allowed to survive a form imposed upon our sensi- 

 bility ? If so, we should have enlarged the prison in which this sensi- 

 bility is confined, but it would always be a prison. 



This continuum has a certain number of properties, exempt from 

 all idea of measurement. The study of these properties is the object 

 of a science which has been cultivated by many great geometers and 

 in particular by Eiemann and Betti and which has received the name 

 of analysis situs. In this science abstraction is made of every quan- 

 titative idea and, for example, if we ascertain that on a line the point 

 B is between the points A and C, we shall be content with this ascer- 

 tainment and shall not trouble to know whether the line ABC is straight 

 or curved, nor whether the length AB is equal to the length BC, or 

 whether it is twice as great. 



The theorems of analysis situs have, therefore, this peculiarity that 

 they would remain true if the figures were copied by an inexpert 

 draftsman who should grossly change all the proportions and replace 

 the straights by lines more or less sinuous. In mathematical terms, 

 they are not altered by any l point-transformation ' whatsoever. It 

 has often been said that metric geometry was quantitative, while 

 projective geometry was purely qualitative. That is not altogether 

 true. The straight is still distinguished from other lines by properties 

 which remain quantitative in some respects. The real qualitative 

 geometry is, therefore, analysis situs. 



The same questions which came up apropos of the truths of 

 Euclidean geometry, come up anew apropos of the theorems of analysis 

 situs. Are they obtainable by deductive reasoning? Are they dis- 

 guised conventions? Are they experimental verities? Are they the 



VOL. LXIX. — 26. 



